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Register a new userGenerating Pareto records
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Added by
James Allen Fill
Publication date
2019
fields
Informatics Engineering
and research's language is
English
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We present, (partially) analyze, and apply an efficient algorithm for the simulation of multivariate Pareto records. A key role is played by minima of the record-setting region (we call these generators) each time a new record is generated, and two highlights of our work are (i) efficient dynamic maintenance of the set of generators and (ii) asymptotic analysis of the expected number of generators at each time.
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For iid $d$-dimensional observations $X^{(1)}, X^{(2)}, ldots$ with independent Exponential$(1)$ coordinates, consider the boundary (relative to the closed positive orthant), or frontier, $F_n$ of the closed Pareto record-setting (RS) region [ mbox{RS}_n := {0 leq x in {mathbb R}^d: x otprec X^{(i)} mbox{for all $1 leq i leq n$}} ] at time $n$, where $0 leq x$ means that $0 leq x_j$ for $1 leq j leq d$ and $x prec y$ means that $x_j < y_j$ for $1 leq j leq d$. With $x_+ := sum_{j = 1}^d x_j$, let [ F_n^- := min{x_+: x in F_n} quad mbox{and} quad F_n^+ := max{x_+: x in F_n}, ] and define the width of $F_n$ as [ W_n := F_n^+ - F_n^-. ] We describe typical and almost sure behavior of the processes $F^+$, $F^-$, and $W$. In particular, we show that $F^+_n sim ln n sim F^-_n$ almost surely and that $W_n / ln ln n$ converges in probability to $d - 1$; and for $d geq 2$ we show that, almost surely, the set of limit points of the sequence $W_n / ln ln n$ is the interval $[d - 1, d]$. We also obtain modifications of our results that are important in connection with efficient simulation of Pareto records. Let $T_m$ denote the time that the $m$th record is set. We show that $F^+_{T_m} sim (d! m)^{1/d} sim F^-_{T_m}$ almost surely and that $W_{T_m} / ln m$ converges in probability to $1 - d^{-1}$; and for $d geq 2$ we show that, almost surely, the sequence $W_{T_m} / ln m$ has $liminf$ equal to $1 - d^{-1}$ and $limsup$ equal to $1$.
We establish a fundamental property of bivariate Pareto records for independent observations uniformly distributed in the unit square. We prove that the asymptotic conditional distribution of the number of records broken by an observation given that the observation sets a record is Geometric with parameter 1/2.
This paper leverages machine-learned predictions to design competitive algorithms for online conversion problems with the goal of improving the competitive ratio when predictions are accurate (i.e., consistency), while also guaranteeing a worst-case competitive ratio regardless of the prediction quality (i.e., robustness). We unify the algorithmic design of both integral and fractional conversion problems, which are also known as the 1-max-search and one-way trading problems, into a class of online threshold-based algorithms (OTA). By incorporating predictions into design of OTA, we achieve the Pareto-optimal trade-off of consistency and robustness, i.e., no online algorithm can achieve a better consistency guarantee given for a robustness guarantee. We demonstrate the performance of OTA using numerical experiments on Bitcoin conversion.
We consider a branching random walk on the lattice, where the branching rates are given by an i.i.d. Pareto random potential. We show that the system of particles, rescaled in an appropriate way, converges in distribution to a scaling limit that is interesting in its own right. We describe the limit object as a growing collection of lilypads built on a Poisson point process in $mathbb{R}^d$. As an application of our main theorem, we show that the maximizer of the system displays the ageing property.
A scheduled arrival process is one in which the n th arrival is scheduled for time n, but instead occurs at a different time. The difference between the scheduled time and the arrival time is called the perturbation. The sequence of perturbations is assumed to be iid. We describe here the behavior of a single server queue fed by such traffic in which the processing times are deterministic. A particular focus is on perturbation with Pareto-like tails but with finite mean. We obtain tail approximations for the steady-state workload in both cases where the queue is critically loaded and under a heavy-traffic regime. A key to our approach is our analysis of the tail behavior of a sum of independent Bernoulli random variables with success probability following a power law with parameter strictly larger than 1.
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